Gauging $\mathbb{Z}_N$ symmetries of Narain CFTs

We investigate the gauging of a $\mathbb{Z}_N$ symmetry in lattice conformal field theories (CFTs), also known as Narain CFTs. For prime $N$, we derive a spin selection rule for operators in a $\mathbb{Z}_N$ charge-twisted sector of a general bosonic CFT. Using this result, we formulate the gauging procedures in lattice CFTs as modifications of the momentum lattices by a lattice vector that specifies a non-anomalous $\mathbb{Z}_N$ symmetry. Applying this formulation to code CFTs, i.e., Narain CFTs constructed from error-correcting codes, we express the torus partition functions of the orbifolded and parafermionized theories in terms of the weight enumerator polynomials of the underlying codes. As an application, we identify a class of codes that yield self-dual bosonic CFTs under the orbifolding by a $\mathbb{Z}_N$ symmetry.